What a prior is, and how conjugacy works

Question

Explain what a prior is and its role in Bayesian inference, work through the Beta-Binomial conjugate update, and describe when the prior matters a lot versus a little.

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What a prior is. In Bayesian inference, the parameter itself has a distribution. The prior \pi(	heta) encodes what you believe about 	heta before seeing data, base rates, economic structure, or plain ignorance. Data enter through the likelihood, and Bayes' rule combines them into the posterior: \pi(	heta \mid x) \propto f(x \mid 	heta)\,\pi(	heta). The prior is the Bayesian's honest input: a channel for outside knowledge, and also the thing critics point to as subjective. Conjugacy: Beta-Binomial. A prior is conjugate to a likelihood if the posterior stays in the same family, which makes…

What a prior is, and how conjugacy works

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